Nash Dynamics in Congestion Games with Similar Resources

We study convergence of ε-Nash dynamics in congestion games when delay functions of all resources are similar. Delay functions are said to be similar if their values are within a polynomial factor at every congestion level. We show that for any ε > 0, an ε-Nash dynamics in symmetric congestion games with similar resources converge in steps polynomial in 1/ε and the number of players are resources, yielding an FPTAS. Our result can be contrasted with that of Chien and Sinclair [3], which showed polynomial convergence result for symmetric congestion games where the delay functions have polynomially bounded jumps. Our assumption of similar delay functions is orthogonal to that of bounded jumps in that neither assumption implies the other. Our convergence result also hold for several natural variants of ε-Nash dynamics, including the most general polynomial liveness dynamics, where each player is given a chance to move frequently enough. We also extend our positive results to give an FPTAS for computing equilibrium in asymmetric games with similar resources, in which players share k distinct strategy spaces for some constant k. We complement our positive results by showing that computing an exact pure Nash equilibrium in symmetric congestion game with similar resources is PLS-complete. Furthermore, we show that for any ε > 0, all sequences of ε-Nash dynamics takes exponential steps to reach an approximate equilibrium in general congestion games with similar resources, as well as in symmetric congestion games with two groups of similar resources.